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# Higher spin dirac operators in two vector variables in Clifford analysis

(2011)
Author
Promoter
(UGent)
Organization
Abstract
The aim of this doctoral thesis is to study the construction and some properties of a speci c higher spin Dirac operator in Cliff ord analysis. In general, a higher spin Dirac operator D is the unique (up to a multiplicative constant) elliptic conformally invariant fi rst-order di fferential operator acting on differentiable functions with values in a vector space V which is an irreducible representation of Spin(m) with a specific highest weight and which can be realised in Clifford analysis as a vector space of polynomials in several vector variables, called the simplicial monogenic polynomials. Higher spin Dirac operators in Cliff ord analysis can be seen as a generalisation of the standard Dirac operator, whose functiontheorical properties lie at the heart of many results in Cliff ord analysis. In its turn, the standard Dirac operator can be seen as the generalisation of the Cauchy-Riemann operator in the complex plane. The above-mentioned speci fic higher spin Dirac operator that plays the main role in this thesis, presents complications that were not yet discovered, even though explicit examples of higher spin Dirac operators have been studied before. Its function-theoretical properties are studied and its homogeneous polynomial null solutions (of a fixed degree) will be discussed. We can reveal right now that the vector space of these null solutions is no longer irreducible with respect to the Spin group, contrary to the case of the standard Dirac operator. It appears that this remains true for other higher spin Dirac operators, which makes the study of these operators complicated. Therefore, an important part of this thesis was to describe the decomposition of this kernel space into Spin(m)-irreducibles summands, which we have labeled by their highest weight. This decomposition is proved by induction. The decomposition of the kernel space into these irreducibles can be visualised in an aesthetically pleasing way. At that point in the thesis it is known how the vector space of h-homogeneous null solutions of our operator behaves under the action of the Spin group. We have tried to solve this problem.
Keywords
Clifford analysis, Higher spin operator

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## Citation

Chicago
Van de Voorde, Liesbet. 2011. “Higher Spin Dirac Operators in Two Vector Variables in Clifford Analysis”. Ghent, Belgium.
APA
Van de Voorde, Liesbet. (2011). Higher spin dirac operators in two vector variables in Clifford analysis. Ghent, Belgium.
Vancouver
1.
Van de Voorde L. Higher spin dirac operators in two vector variables in Clifford analysis. [Ghent, Belgium]; 2011.
MLA
Van de Voorde, Liesbet. “Higher Spin Dirac Operators in Two Vector Variables in Clifford Analysis.” 2011 : n. pag. Print.
```@phdthesis{4107236,
abstract     = {The aim of this doctoral thesis is to study the construction and some properties of a speci\unmatched{000c}c higher spin Dirac operator in Cliff\unmatched{000b}ord analysis. In general, a higher spin Dirac operator D\unmatched{0015} is the unique (up to a multiplicative constant) elliptic conformally invariant fi\unmatched{000c}rst-order di\unmatched{000b}fferential operator acting on differentiable functions with values in a vector space V which is an irreducible representation of Spin(m) with a specific highest weight and which can be realised in Clifford analysis as a vector space of polynomials in several vector variables, called the simplicial monogenic polynomials. Higher spin Dirac operators in Cliff\unmatched{000b}ord analysis can be seen as a generalisation of the standard Dirac operator, whose functiontheorical properties lie at the heart of many results in Cliff\unmatched{000b}ord analysis. In its turn, the standard Dirac operator can be seen as the generalisation of the Cauchy-Riemann operator in the complex plane. The above-mentioned speci\unmatched{000c}fic higher spin Dirac operator that plays the main role in this thesis, presents complications that were not yet discovered, even though explicit examples of higher spin Dirac operators have been studied before. Its function-theoretical properties are studied and its homogeneous polynomial null solutions (of a \unmatched{000c}fixed degree) will be discussed. We can reveal right now that the vector space of these null solutions is no longer irreducible with respect to the Spin group, contrary to the case of the standard Dirac operator. It appears that this remains true for other higher spin Dirac operators, which makes the study of these operators complicated. Therefore, an important part of this thesis was to describe the decomposition of this kernel space into Spin(m)-irreducibles summands, which we have labeled by their highest weight. This decomposition is proved by induction. The decomposition of the kernel space into these irreducibles can be visualised in an aesthetically pleasing way. At that point in the thesis it is known how the vector space of h-homogeneous null solutions of our operator behaves under the action of the Spin group. We have tried to solve this problem.},
author       = {Van de Voorde, Liesbet},
keyword      = {Clifford analysis,Higher spin operator},
language     = {eng},
pages        = {266},
school       = {Ghent University},
title        = {Higher spin dirac operators in two vector variables in Clifford analysis},
year         = {2011},
}

```